Categories & morphisms
Define objects through composable structure-preserving maps rather than by inspecting their internal elements alone.
Side 218Side Studies / Research
Subject
Category Theory
Purpose
Mathematical structures studied through objects, morphisms, functors and universal properties that emphasize relationships over internal representation.
Structure
05 movesFormal systemV0
Definitions → structures → relations → proof → application
Compare structures by their mappings.
Category theory provides a high-level language for recurring patterns across mathematics, but its abstractions are useful only when universal properties clarify concrete constructions.
Functors
Translate between categories while preserving composition and identity, revealing structural analogies.
Natural transformations
Compare functors through coherent families of morphisms rather than isolated correspondences.
Limits & colimits
Express products, pullbacks, quotients and related constructions by universal mapping properties.
Adjunctions
Capture pairs of constructions linked by a natural correspondence of morphisms, often generating broad mathematical dualities.
Keep the boundaries visible.
object ≠ set of elements
isomorphism ≠ equality
abstraction ≠ generality without content
Questions that organize the Side.
What information is preserved when a construction is characterized universally?
Why are natural transformations stronger than pointwise correspondences?
When does categorical language simplify a problem rather than merely rename it?
What should carry weight here?
Formal definitions and universal properties must be checked in concrete categories; abstraction earns its place by proving reusable results or clarifying structure.